Bristol
In Planck's model of the harmonic oscillator (HO) a century ago, both the energy and the phase space were quantized according to epsilon(n) = nh nu, n = 0, 1, 2..., and integral integral dp(x) dx = h. By referring to just these two relations, we show how the adoption of cycle-averaged phase-space states (CAPSSs) leads to the quantum mechanical energy spectrum of the HO, < E(n)> = (n + 1/2)h nu, n = 0, 1, 2,..., where < E(n)> are the average energies, and to < J(n)> = (n + 1/2)h/2 pi, where < J(n)> are the average actions. When anharmonicity to all orders is added in the form of the Morse oscillator (MO), the concept of CAPSS is implemented in terms of action-angle variables and it is shown that the use of < J(n)> of each MO CAPSS also produces the correct discrete spectrum of the MO, again without applying quantum mechanics (QM). In addition, the concept of CAPSS leads to two well-known post-QM relations which are obtained in terms of time averages of the classical trajectories and of < J(n)> : (1) closed integral p dx = 2 pi < J(n)> = (n + 1/2)h, which is the quantum condition of the old quantum theory, albeit with half-integers (i.e. the result of the WKB approximation), and (2) (Delta p Delta x)(n) >= h/4 pi, which is the Heisenberg uncertainty principle. It is shown via numerical computations that, for two MOs, one with intermediate anharmonicity, supporting 22 levels, and another with strong anharmonicity, with 5 levels, the quantities(Delta p Delta x)(n), n = 0, 1,..., 4, which are computed classically for the appropriately chosen trajectories agree very well with the results of computations that apply QM. The introduction of the CAPSS and the concomitant results underline the significance of the concept of the state of the system in physics, both classical and quantum.
National Hellenic Research Foundation (NHRF)
